Method and apparatus for generating a specific flip angle distribution in mri

ABSTRACT

The present invention provides a method and apparatus for generating a specific flip angle distribution in magnetic resonance imaging; the method uses a plurality of RF transmission coils combined with linear and nonlinear spatial encoding magnetic fields to generate a homogeneous flip angle distribution.

RELATED APPLICATIONS

This application claims priority to Taiwan Application Serial Number102149210, filed on Dec. 31, 2013, which is herein incorporated byreference.

BACKGROUND

1. Technical Field

The present disclosure relates to magnetic resonance imaging (MRI); moreparticularly, the present disclosure relates to method and apparatus forgenerating a flip angle distribution in MRI.

2. Description of Related Art

High-field MRI offers a great promise to generate images with highsignal-to-noise ratio (SNR). Yet the major technical challenge is theinhomogeneous flip angle distribution when a volume RF coil is used forRF excitation. This artifact is due to the deleterious interactionbetween the dielectric properties of the sample and the radio-frequencyfield; consequently, when an object with the size approximating to thehuman head is imaged by high-field (>=3T) MRI, the flip angledistribution is spatially varying, where typically a larger flip angleat the center of the field-of-view (FOV) and a smaller flip angle at theperiphery of the FOV. This causes images with a spatially dependent Ticontrast, which makes clinical diagnosis difficult.

Different methods for mitigating B₁ ⁺ inhomogeneity have been proposed;for example, dedicated volume radio-frequency (RF) coils have beendesigned for use in high-field MRI; another method is using spatiallyselective RF excitation, wherein spatially selective RF excitationdesigns RF and gradient waveforms to form an inhomogeneous B₁ ⁺ field ina volume coil, and finally generates a more homogeneous flip angledistribution. Alternatively, it has been suggested that flip angledistribution can become more homogeneous by using simultaneous RFexcitation from multiple RF coils, such as RF shimming and transmitSENSE. Notably, parallel RF transmission (pTx) methods allow higherdegree of freedom in RF pulse and gradient waveform design than RFshimming, because different RF pulse waveforms can be delivered to eachRF coil independently; however, the challenges of parallel RFtransmission method include the complexity of the RF electronics andcoil construction in order to achieve simultaneous excitation, thenecessity of accurate estimates of phases and amplitudes of the B₁ ⁺maps for each RF coil, and the specific absorption rate (SAR)management.

Recently, it has been demonstrated that nonlinear spatial encodingmagnetic fields (SEMs) can be used in MRI spatial encoding in order toimprove spatiotemporal resolution; preliminary studies using quadraticnonlinear SEMs for RF excitation and small FOV imaging have beenreported. Nonlinear spatial encoding magnetic fields can also be used tomitigate the inhomogeneity of the flip angle distribution; under thesmall flip angle approximation, there are theories indicating how thespatial distribution of the flip angle is controlled by time-varyinglinear and nonlinear spatial encoding magnetic fields and RF pulsewaveforms.

In summary, different techniques have been broached to improve theuniformity of the flip angle spatial distribution in high field MRI;however, at present time, there is no such technique which combines RFshimming and the usage of linear and nonlinear spatial encoding magneticfields to achieve a homogeneous flip angle distribution.

SUMMARY

Under the small flip angle approximation, the present inventionincorporates RF shimming technique and the method of remapping the B₁ ⁺map into a lower dimension coordinate system. If the remapping issuccessful, the iso-intensity contours of the SEMs are similar to theiso-intensity contours of B₁ ⁺ field, which helps pulse sequence designusing linear and nonlinear SEMs to achieve a homogeneous flip angledistribution. The present invention incorporates RF shimming techniquewith the technique of linear and nonlinear SEMs; without the constraintof using parallel transmission techniques, a RF excitation with spatialselectivity can be accomplished effectively, and a homogeneous flipangle distribution can be achieved.

The present invention provides a method for generating a flip anglespatial distribution of magnetic resonance imaging, comprising:incorporating the usage of one or a plurality of spatial encodingmagnetic fields and the usage of a plurality of RF coils to generate aspecific spatial distribution of flip angle, wherein the plurality of RFcoils excite a B₁ ⁺ field jointly, the ratio of signal amplitudes of theplurality of RF coils is kept invariant during the excitation process,and the phase relationship of signals from the plurality of RF coils iskept invariant during the excitation process.

Preferably, in the aforementioned method for generating a flip anglespatial distribution of magnetic resonance imaging, the plurality ofspatial encoding magnetic fields include linear and nonlinear spatialencoding magnetic fields.

Preferably, in the aforementioned method for generating a flip anglespatial distribution of magnetic resonance imaging, the generatedspatial distribution of flip angle is substantially homogeneous.

Preferably, the aforementioned method for generating a flip anglespatial distribution of magnetic resonance imaging further comprisingadjusting the signal amplitudes of the plurality of RF coils by a singlecontroller, wherein the ratio of signal amplitudes and the phaserelationship of signals of the plurality of RF coils are kept invariantduring the excitation process.

In another aspect, the present invention provides an apparatus forgenerating a flip angle spatial distribution of magnetic resonanceimaging, comprising: one or a plurality of spatial encoding magneticfield coils; and a plurality of RF coils incorporated with the one orplurality of spatial encoding magnetic field coils to generate aspecific spatial distribution of flip angle, wherein the plurality of RFcoils excite a B₁ ⁺ field jointly, the ratio of signal amplitudes of theplurality of RF coils is kept invariant during the excitation process,and the phase relationship of signals from the plurality of RF coils iskept invariant during the excitation process.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosure can be more fully understood by reading the followingdetailed description of the embodiments, with reference made to theaccompanying drawings as follows:

FIG. 1A shows the B₁ ⁺ amplitude and phase maps of 8 transmit channelsand the CP mode combination in a saline phantom

FIG. 1B shows the B₁ ⁺ amplitude and phase maps of 8 transmit channelsand the CP mode combination in a human head.

FIG. 2 shows the real and imaginary parts of the B₁ ⁺ maps before andafter total variation (TV) de-noising of channel 1 using human headdata.

FIG. 3 shows the spatial distribution of B₁ ⁺ using RF shimming withoriginal 8 B₁ ⁺t maps from 8 transmit coils in a human head (left), RFshimming with TV denoised 8 B₁ ⁺t maps from 8 transmit coils in a humanhead (middle), and the difference of the distribution of the B₁ ⁺ mapbetween these two (right).

FIG. 4 shows the distribution of B_(1shim) ⁺(x,y) in the conventional RFshimming method (using only linear SEMs; top row), the optimized 2-spokecombined SAGS and RF shimming method (middle row), and the optimized4-spoke combined SAGS and RF shimming method (bottom row) usingsimulations, experimental saline phantom data, and experimental humanhead data.

FIG. 5 shows the relationship between h(x,y) and 1/B_(1shim) ⁺(x,y) inthe conventional RF shimming method (using only linear SEMs; top row),the optimized 2-spoke combined SAGS and RF shimming method (middle row),and the optimized 4-spoke combined SAGS and RF shimming method (bottomrow) using simulations, experimental saline phantom data, andexperimental human head data. The mapping error quantified by theroot-mean-square of the residuals between (h(x,y), 1/B_(1shim) ⁺(x,y))pairs and a fitted curve using 10^(th)-order polynomials is reported foreach plot.

FIG. 6 shows the flip angle distribution maps obtained with theconventional RF shimming method (using only linear SEMs), the optimized2-spoke combined SAGS and RF shimming method, the optimized 4-spokecombined SAGS and RF shimming method, and the combined SAGS and RFshimming method with 99 equi-spaced spokes using simulations,experimental saline phantom data, and experimental human head data. Therelative standard deviation σ is reported below each map.

FIG. 7 shows the flip angle distribution maps generated by the optimized2-spoke combined SAGS and RF shimming method using simulations,experimental saline phantom data, and experimental human head data. Theinitial guess for B_(1shim) ⁺(x,y) was calculated by the conventional RFshimming method using only linear SEMs. The relative standard deviationσ is reported below each plot.

DETAILED DESCRIPTION Theory

A. Spatially Selective RF Excitation with Generalized Spatial EncodingMagnetic Fields (SAGS) Under the Small Flip Angle ApproximationTo allow the complete description of combined SAGS and RF-shimming, wefirst introduce the necessary theoretical framework. For an MRI systemwith n distinct configurations of SEMs turning on during RF excitation,we use the dimensionless variable f(r)=[f₁(r), . . . , f_(n)(r)] todescribe the spatial distributions of the z-components of these SEMs. Tofacilitate the description of the arbitrary spatial distribution off(r), we define the maximal and the minimal values among all componentsof f(r) within the imaging object to be 1 and 0, respectively.g(t)=[g₁(t) . . . g_(n)(t)] describes the instantaneous magnetic fieldstrength of each individual SEM. Accordingly, each component of g(t)clearly defines the instantaneous difference between the minimal andmaximal z-component of the magnetic field generated by each SEM withinthe imaging object. The instantaneous additional z-component of themagnetic field at location r is thus the inner product g(t)·f(r). Herewe assume that the RF transmit field B₁ ⁺(r,t) is spatiotemporallyseparable: B₁ ⁺(r,t)=/B_(1,r) ⁺(r) B_(1,t) ⁺(t), where B_(1,t) ⁺(t) is awaveform of the RF transmit field, and B_(1,r) ⁺(r) is a spatialdistribution of the ratio between B₁ ⁺(r,t) and B₁ ⁺(t).

Taking the small flip angle approximation (1) and assuming the initialmagnetization [M_(x)(r,0), M_(y)(r,0), M_(z)(r,0)]^(T)=[0, 0, 1]^(T),the spatial distribution of the transverse magnetization M_(xy)(r), theRF pulse waveform B_(1t) ⁺(t), and the temporal integral of SEMs(linear/nonlinear MRI gradients) over time are related to each other byan inverse Fourier transform (2-4)

$\begin{matrix}{{{M_{xy}(r)} = {\int_{K}^{\;}{{W(k)}{B_{1\; r}^{+}(r)}{\exp \left\lbrack {j\; 2\; \pi \; {{f(r)} \cdot k}} \right\rbrack}{\delta (k)}\ {k}}}}{{W(k)} = {{W\left( {k(t)} \right)} = \frac{j\; \gamma \; {B_{1\; t}^{+}(t)}}{{k^{\prime}(t)}}}}{{k(t)} = {\left\lbrack {{k_{1}(t)},\ldots \mspace{14mu},{k_{n}(t)}} \right\rbrack = {\gamma {\int_{t}^{T}{{g(u)}\ {{u}.}}}}}}} & \lbrack 1\rbrack\end{matrix}$

Note that the notation k(t) in this study is different from that inconventional MRI (1): we chose k(t) to express the maximal phasedifference of the transverse magnetization precession within the imagingobject at time instant t. Since a k-space trajectory has the one-to-onecorrespondence between k(t) and t, we omit the t argument in k(t) anduse k in the following. Additionally, we use a delta function δ(k),which is nonvanishing at all excited k-space points, to describe ak-space trajectory.

To achieve a practical slice-selective RF excitation, we propose to usea spoke k-space trajectory (5), with spokes locate at k_(s) and s∈S,where S denotes the set including all spokes. Without losing generality,we consider that the slices are distributed over the z-axis and thatonly the central slice (z=0) is excited. Accordingly, Eq. [1] in a spokek-space trajectory becomes

$\begin{matrix}{{{M_{xy}\left( {x,y} \right)} = {{B_{1\; r}^{+}\left( {x,y} \right)}{\sum\limits_{s \in S}^{\;}\; {{W_{F}\left( k_{s} \right)}{\exp \left\lbrack {2\; \pi \; j\; {k_{s} \cdot {f\left( {x,y} \right)}}} \right\rbrack}}}}},} & \lbrack 2\rbrack\end{matrix}$

where M_(xy)(x,y) is the distribution of the excited transversemagnetization at z=0 plane. It should be noted that M_(xy)(x,y)/B₁⁺(x,y) and f(x,y)=(x,y) are related by a 2D Fourier transform. Naturallysuch a design is on a 2D space.

SAGS seeks to remap M_(xy)(x,y)/B₁ ⁺(x,y) by a new variable h(x,y) withdimension lower than that of f(x,y) (6). Using a spoke trajectory andconsidering a 2D slice-selective excitation scheme, the range of h(x,y)is one dimensional: h(x,y)=h(x,y). If this remapping is possible, thenEq. [2] becomes

$\begin{matrix}{{{{M_{xy}\left( {x,y} \right)}/{B_{1}^{+}\left( {x,y} \right)}} \approx {{\overset{\sim}{M}}_{xy}\left( {h\left( {x,y} \right)} \right)}} = {\sum\limits_{s \in S}^{\;}\; {W_{F}\left( k_{s} \right){{\exp \left\lbrack {2\; \pi \; j\; {k_{s} \cdot {f\left( {x,y} \right)}}} \right\rbrack}.}}}} & \lbrack 3\rbrack\end{matrix}$

This shows that remapping can be advantageous if {tilde over(M)}_(xy)(h(x,y)) can be represented by a few spokes using a onedimensional variable h(x,y). Specifically, Eq. [3] shows thatM_(xy)(x,y)/B₁ ⁺(x,y) and h(x,y) are now related by a 1D Fouriertransform. Accordingly, such a design is now on a 1D space. Reducing thek-space dimension after B₁ ⁺ remapping implies that a shorter k-spacetrajectory can be used to achieve a similar distribution of thetransverse magnetization.

B. Combined SAGS and RF Shimming

When RF shimming hardware is available, B₁ ⁺(x,y) in Eq. [3] can be anylinear combination of the B₁ ⁺ maps of the multiple RF coils. Thus wemay be able to find one linear combination of B₁ ⁺ maps

$\begin{matrix}{{B_{1\; {comb}}^{+}\left( {x,y} \right)} = {\sum\limits_{c = 1}^{n_{c}}\; {\xi_{c}{B_{1}^{c}\left( {x,y} \right)}}}} & \lbrack 4\rbrack\end{matrix}$

such that M_(xy)(x,y)/B_(1comb) ⁺(x,y) can be better approximated by{tilde over (M)}_(xy)(h(x,y)). Here n_(c) denotes the number of RFcoils. B₁ ^(c)(x,y) denotes the B₁ ⁺ map of RF coil channel c.

The combined SAGS and RF shimming method needs to find the remappingvariable h(x,y), the remapping function {tilde over (M)}_(x,y)(•), andξ_(c) coefficients for B_(1comb) ⁺(x,y), such that the followingapproximation holds:

M _(xy)(x,y)/B _(1comb) ⁺(x,y)≈{tilde over (M)} _(xy)(h(x,y))  [5]

C. Homogeneous Flip Angle Excitation

When aiming at achieving a homogeneous flip angle distribution, we onlyconsidered the magnitude of the combined B₁ ⁺, since the phasedistribution is generally not important in most clinical applications.

$\begin{matrix}{{B_{1\; {shim}}^{+}\left( {x,y} \right)} = {{\sum\limits_{c = 1}^{n_{c}}{\xi_{c}{B_{1}^{c}\left( {x,y} \right)}}}}} & \lbrack 6\rbrack\end{matrix}$

B_(1shim) ⁺(x,y), is the absolute value of the combined B₁ ⁺ maps, i.e.,B_(1comb) ⁺(x,y), in Eq. [4]. In this study, we also assumed that onlySEMs of polynomial order 2 or lower could be used:

$\begin{matrix}{{h\left( {x,y} \right)} = {\sum\limits_{q = 0}^{2}{\sum\limits_{r = 0}^{q}{{\upsilon_{q,r}\left( {x^{r},y^{q - r}} \right)}.}}}} & \lbrack 7\rbrack\end{matrix}$

Using m_(xy) to denote the desired spatial distribution of transversemagnetization, we assumed that m_(xy)/B_(1shim) ⁺(x,y) is spatiallysmooth. Considering in our case that m_(xy)/B_(1shim) ⁺(x,y) is realvalued, we could arbitrarily use a linear combination of p cosinefunctions to approximate {tilde over (M)}_(xy)(•) parameterized by h:

$\begin{matrix}{{{\overset{\sim}{M}}_{xy}(h)} = {\sum\limits_{q = 0}^{p - 1}{\kappa_{q}{{\cos ({qh})}.}}}} & \lbrack 8\rbrack\end{matrix}$

The reasons we chose cosine functions are: 1) cosine functions are goodbases when there is a sufficient number of harmonics, 2) cosinefunctions are related to even number of spokes located in conjugatedlocations in k-space. For example, when only q=1 is used in equation[8], {tilde over (M)} can be realized by two spokes, which aresymmetrically located around the k-space origin with equal amplitudesand conjugate phases.

In summary, our proposed pulse design is an optimization problem aimingat adjusting parameters {κ, ξ, ν} to minimize the following error:

$\begin{matrix}{\left\{ {\kappa^{opt},\xi^{opt},v^{opt}} \right\} = {\underset{\kappa,\xi,v}{\arg \; \min}{{1 - {{\sum\limits_{c = 1}^{n_{c}}{\xi_{c}{B_{1}^{c}\left( {x,y} \right)}{\sum\limits_{s = 0}^{p - 1}{\kappa_{s}{\cos\left( {\sum\limits_{q = 0}^{2}{\sum\limits_{r = 0}^{q}{v_{q,r}\left( {x^{r}y^{q - r}} \right)}}} \right)}}}}}}}}_{2}^{2}}} & \lbrack 9\rbrack\end{matrix}$

After we obtain the remapping coefficients {ξ^(opt), ν^(opt)}, we solvethe optimization problem to determine the spoke locations in thek_(z)-k_(h) space, which are symmetrically distributed around thek-space center, and spoke amplitudes to achieve a homogeneous flip angledistribution:

$\begin{matrix}{\left\{ {\alpha^{opt},\beta^{opt}} \right\} = {\underset{\alpha,\beta}{\arg \; \min}{{m_{xy} - {{\beta_{1{shim}}\left( {x,y} \right)}{\sum\limits_{s = 1}^{S}{\alpha_{s}{\exp \left( {{j\beta}_{s}{h\left( {x,y} \right)}} \right)}}}}}}_{2}^{2}}} & \lbrack 10\rbrack\end{matrix}$

D. Algorithm

We used the following algorithm to derive all parameters {^(opt),^(opt), ^(opt), ^(opt)}, separating the pulse design process into twoparts.

Part I

The first part aims to solve the optimization problem defined in Eq.[9].

Step 1. Initialization

We first assume that {tilde over (M)}_(xy) (•) can be approximated byone cosine function. Our chosen initial guess for ξ_(c) is thecircular-polarized (CP) mode (7), and the initial guess for κ_(p) is

$\kappa_{p}^{old} = \left\{ {\begin{matrix}{{\max \left\{ {1/{B_{1{shim}}^{+}\left( {x,y} \right)}} \right\}},} & {p = 1} \\{0,} & {otherwise}\end{matrix}.} \right.$

Accordingly,

$v = {\underset{\upsilon_{q,r}}{\arg \; \min}{{{{\sum\limits_{q = 0}^{2}{\sum\limits_{r = 0}^{q}{\upsilon_{q,r}\left( {x^{r},y^{q - r}} \right)}}} - {{arc}\; \cos \; \left( \frac{1}{\alpha_{1}{B_{1{shim}}^{+}\left( {x,y} \right)}} \right)}}}_{2}^{2}.}}$

We also define B_(1phase) ^(+old)(x,y) as the phase distribution of

$\sum\limits_{c = 1}^{n_{c}}{\xi_{c}^{old}{{B_{1}^{c}\left( {x,y} \right)}.}}$

Step 2. Iterative Updating

Using a combination of the gradient descent algorithm and least squaressolution iteratively, we adjust { } to minimize

${{\Phi \left( {\kappa_{1},\gamma,\upsilon,{B_{1{phase}}^{+}\left( {x,y} \right)}} \right)} = {{^{j\; {B_{1{phase}}^{+}{({x,y})}}} - {{\kappa_{1}\left( {\sum\limits_{c = 1}^{n_{c}}{\xi_{c}{B_{1}^{c}\left( {x,y} \right)}}} \right)}{\cos \left( {\sum\limits_{q = 0}^{2}{\sum\limits_{r = 0}^{q}{\upsilon_{q,r}\left( {x^{r},y^{q - r}} \right)}}} \right)}}}}_{2}^{2}},$

where B_(1phase) ⁺(x,y) denotes the phase distribution of

$\sum\limits_{c = 1}^{n_{c}}{\xi_{c}{{B_{1}^{c}\left( {x,y} \right)}.}}$

Specifically, Step 2.1:

With given {κ^(old), ξ^(old), υ^(old), B_(1phase) ^(+old)(x,y)} weupdated the value of υ_(q,r) using the gradient descent algorithm withthe step size λ:υ^(new)=υ^(old)−λ∇_(v)Φ. Note if Φ(κ^(old), ξ^(old), υ^(new), B_(1phase)^(+old)(x,y))≧Φ(κ^(old), ξ^(old), υ^(old), B_(1phase) ^(+old)(x,y)),λmust be reduced by half and Step 2.1 should be repeated.

Step 2.2:

Use a least squares algorithm to find the new κ₁:

$\kappa_{1}^{new} = {\underset{\alpha_{1}}{\arg \; \min}\left\{ {\Phi \left( {\kappa_{1},\xi^{old},\upsilon^{new},{B_{1{phase}}^{+ {old}}\left( {x,y} \right)}} \right)} \right\}}$

Step 2.3:

Use a least squares algorithm to find the new:

$\xi^{new} = {\underset{\gamma}{\arg \; \min}{\left\{ {\Phi \left( {\kappa_{1}^{new},\xi^{old},\upsilon^{new},{B_{1{phase}}^{+ {old}}\left( {x,y} \right)}} \right)} \right\}.}}$

Update B_(1phase) ⁺(x,y) using ξ^(new): B_(1phase) ^(+new)(x,y) thephase distribution of

$\sum\limits_{c = 1}^{n_{c}}{\xi_{c}^{new}{{B_{1}^{c}\left( {x,y} \right)}.}}$

We repeat Step 2 until the cost converges: Φ(κ^(old), ξ^(old), υ^(old),B_(1phase) ^(+old)(x,y))−Φ(κ^(old), ξ^(old), υ^(new), B_(1phase)^(+old)(x,y))<∈. When not converging, we update {κ₁ ^(old), ξ^(old),υ^(old), B_(1phase) ^(+old)(x,y)}←{κ₁ ^(new), ξ^(new), υ^(new),B_(1phase) ^(+new)(x,y)} and repeat Step 2.1 to Step 2.3. Atconvergence, we obtain the optimized parameters {κ₁ ^(opt), ξ^(opt),υ^(opt)}←{κ₁ ^(new), ξ^(new), υ^(new)}. This gives us the optimal designwith 2 spokes.

For a trajectory using more than 2 spokes, we first optimize two spokesusing steps 1 and 2 to obtain {κ₁ ^(opt), ξ^(opt), υ^(opt)}. This givesus the initial guess for {κ₁ ^(opt), ξ^(opt), υ^(opt)}. Then, we use thesame procedure in step 2 to obtain {κ₁ ^(opt), ξ^(opt), υ^(opt)}, exceptthat now in step 2.2 all κ_(p) should be updated.

Part II.

The second part of the pulse design uses the estimated B_(1shim) ⁺(x,y)and h(x,y) to find spoke locations and strengths (Eq. 10). It should benoted that the optimization in Part I provides optimized κ_(p)corresponding to locations and amplitudes of spokes. In Part II of ouralgorithm, we seek to further reduce the error in Eq. [9] by using thesame number of spokes in different locations or to approximate thesimilar error by using fewer spokes. To do this, we use the optimized{ξ^(opt), υ^(opt)} to exhaustively search all possible locations β_(s),which are symmetrically located around 0. The amplitudes for these 2Sspokes are expressed by α_(s) The optimization for these 2S spokes nowbecomes

$\left\{ {\alpha,\beta} \right\} = {{\underset{\alpha,\beta}{\arg \; \min}{{m_{xy} - {\beta_{1{shim}}^{+}{\sum\limits_{s = 1}^{S}{\alpha_{s}{\exp \left( {{j\beta}_{s}{h\left( {x,y} \right)}} \right)}}}}}}_{2}^{2}} + {\lambda {{\alpha }_{2}^{2}.}}}$

We added a regularization term ∥α∥₂ ² in the cost function to suppresssolutions requiring excessive peak power.

1st Embodiment Comparison Between RF Shimming with B₁ ⁺ Remapping andConventional RF Shimming

In the first embodiment of the present invention, the sample was modeledas a uniform sphere with average brain electrical properties at 7 T(dielectric constant ∈_(r)=52, electric conductivity σ=0.55 S/m). Thefull-wave electromagnetic (EM) field produced by a circular surface coiladjacent to the homogeneous sphere was expressed in the form of asemi-analytical multipole expansion (8,9). The EM field can beappropriately rotated to express the EM field of any identical circularcoil at a different position near the surface of the sphere, withrespect to the same reference frame of the sample (9). We modeled a20-element transmit array of identical circular coils uniformly packedaround the sphere and for each coil we calculated the magnetic field(B₁) for a uniform grid of voxels (32×32) on a transverse FOV throughthe center of the object. Simulated B₁ ⁺ maps were computed for eachelement of the transmit arrays as B₁ ⁺(r)=B_(1x)(r)+i B_(1y)(r), where iis the imaginary unit and r is the voxel position. All calculations wereimplemented using MATLAB (Mathworks, Natick, USA) on a standard PC.

2nd Embodiment Comparison of the Homogeneity of Transverse Magnetization|M_(xy)| Relative to RF Power from Different Techniques

In the second embodiment of the present invention, both phantom andin-vivo B₁ ⁺ maps were obtained using a Siemens 7T Magnetom scannerequipped with an 8-channel pTX-setup (Siemens, Erlangen, Germany). Forthis purpose a custom 8-element transceive head coil array was used(10). First the Actual Flip-angle Imaging (AFI) method (11) was used toobtain a quantitative B₁ ⁺ map corresponding to the CP mode (3 mmisotropic resolution, TR1/TR2=30/150 ms, TE=1.5 ms, max flip-angle +/−90deg). Subsequently a multi-slice Fast Low Angle Shot (FLASH) sequencewas used to measure the relative signal amplitudes and phasescorresponding to each of the individual channels and CP-mode (3 mmisotropic resolution, TR=500 ms, TE=2.0 ms max flip-angle +/−10 deg).Finally, using the CP-mode, one additional FLASH image was obtained witha longer TE=2.5 ms. Combining both CP-mode FLASH images a ΔB₀ wasconstructed. The quantitative B₁ ⁺ maps corresponding to each of theindividual transmit-channels were derived as described in (12). The samesequence parameters were used for both phantom and in-vivo measurements.Informed consent was obtained from each volunteer, in accordance withthe regulations of our institution.

3rd Embodiment

The noise in estimating the B₁ ⁺ from the empirical data can be observedas some sharp variations in the B₁ ⁺ maps. To reduce noise, we used atotal variation (TV) denoising method (13). The RF shimming coefficientsfor generating a homogenous B₁ ⁺ distribution were calculated based onthe magnitude least square method (14) using CP mode as the initialguess. We also compared RF-shim using original B₁ ⁺ and the denoised B₁⁺ to verify that the denoising process will not cause extra artifact.

In all cases, for both simulated and experimental data, we designedpulses to achieve a homogenous 10° flip angle distribution. In practice,we only searched the optimal solutions for 2 and 4 spokes located atconjugate locations. After optimizing pulse sequence design, the flipangle distribution was calculated based on the Bloch equations. Theperformance of |B₁ ⁺| mitigation was evaluated by the relative standarddeviation σ (15):

σ=std(|M _(xy)|)/mean(|M _(xy)|),  [10]

where std(•) and mean(•) indicate the standard deviation and the mean ofthe transverse magnetization, respectively. Note that σ is a constantwhen the flip angle is small, because the standard deviation and mean ofM_(xy) are linearly proportional to each other.

To evaluate B₁ ⁺ remapping, we plotted pairs (h(x,y),1/B_(1shim) ⁺(x,y))for all voxels inside the imaging object. In the case of perfectremapping (Eq. [3]]), all pairs should be represented by one curve (h).To quantify the accuracy of B₁ ⁺ remapping, we estimated a 10^(th)-orderpolynomial based on all (h(x,y),1/B_(1shim) ⁺(x,y)) and calculated theerrors between data pairs and the fitted curve. The error in B₁ ⁺remapping was quantified as the std(1/B_(1shim)⁺(x,y)−(h(x,y)))/mean((h(x,y))). For comparison, we also simulated theflip angle distribution using the fast k_(z) method (5) and the tailoredexcitation method (15) using conventional RF-shim, which aims atachieving a homogeneous B₁ ⁺ distribution. This should be distinguishedfrom the RF shimming used in our SAGS method, for which we aim atachieving a homogeneous M_(xy) without constraining the RF shim toachieve a homogeneous B₁ ⁺ distribution directly. In other words, weallow RF shim to achieve a tailored B₁ ⁺ distribution, which will beused by SAGS to achieve a homogeneous M_(xy) distribution. To estimatethe energy deposition associated with each method, for the simulateddata we computed the global specific absorption rate (SAR) for eachexcitation, using a previously published method (9). For experimentaldata, we listed the relative amplitudes of spoke at each transmit coilcomparing to the CP mode, such that they generated the same average |B₁⁺|. To make sure the number of spokes number was sufficient in ouralgorithm, we simulated 99 equi-spaced spokes SAGS, which uses the sameB_(1shim) ⁺(x,y) and h(x,y) as 4 spokes SAGS. We also simulated the flipangle distribution using 4 spokes SAGS with the outcome of aconventional RF shim as the initial guess in our algorithm step 1 toinvestigate how the initial guess affects the optimization.

FIG. 1 shows the measured in vivo B₁ ⁺ maps of the brain and phantommeasured at 7 T. FIG. 2 shows one representative channel of the in vivoB₁ ⁺ map before and after TV denoise processing. It can been seen thatboth real and imaginary parts were spatially smooth after denoising.FIG. 3 shows the distribution of B₁ ⁺ map after applying conventional RFshimming, where B₁ ⁺ maps were with and without TV denoising processing.We observed that the difference between two RF shims using original anddenoised B₁ ⁺ maps is moderate. In particular, the sharp magnitudediscontinuity exists in both cases, potentially due to B₁ ⁺ estimationerror. In the subsequent analyses, we used de-noised B₁ ⁺ maps for B₁ ⁺remapping and RF shimming design because we expect spatially smooth B₁ ⁺maps based on electromagnetic theory and we want to avoid the error inassessing remapping accuracy.

FIG. 4 shows B₁ ⁺ maps obtained with the conventional RF shimmingmethod, which aims at achieving the most homogeneous |B₁ ⁺| (top row)and with the combination of SAGS and RF shimming using 2 (middle row)and 4 (bottom row) spokes, for simulated transmit coil sensitivies (leftcolumn) and experimental transmit coil sensitivies, for both a phantom(middle column) and a human head (right column). Regardless of themethod, we observed some rapid intensity changes in the |B₁ ⁺| maps,potentially due to the rapid change of phase of the individual B₁ ⁺ inmeasurements. On visual inspection, conventional RF shimming aloneyielded relatively homogeneous |B₁ ⁺| distribution for both simulationand experimental data. The SAGS method combined with RF shimmingresulted in more spatially inhomogeneous, but smoother |B₁ ⁺|distributions. If this B₁ ⁺ distribution has iso-strength contoursbetter fitted to a quadratic function, we can use nonlinear SEMs andSAGS to reduce the k-space dimension and a homogeneous transversemagnetization distribution can be more efficiently achieved (to bedemonstrated in FIG. 6).

One key requirement for our method to generate homogeneous M_(xy) isfinding a one-dimensional function between 1/B_(1shim) ⁺(x,y) and h(x,y)(Eq. [3]). To validate this requirement after RF shim, we plotted thedistribution between 1/B_(1shim) ⁺(x,y) and h(x,y) at all voxels in theimaging object (FIG. 5). Ideally, this distribution should berepresented by one curve, which was estimated by a 10^(th)-orderpolynomial (red curves in FIG. 5). Comparing between B₁ ⁺ remappingusing B_(1shim) ⁺ and conventional RF shim, across simulations andexperimental data, we found 1/B_(1shim) ⁺(x,y) and h(x,y) are moreclosely related to each other by a smooth one-dimensional function,because the fitted red curves fitted to the distribution of(h(x,y),1/B_(1shim) ⁺(x,y)) pairs. However, there were some outlier datapoints, which might be due to inaccuracy in the experimental B₁ ⁺measurements, as mentioned above. Combining SAGS and RF shimming with 4spokes rather than two improved the mapping between 1/B_(1shim) ⁺(x,y)and h(x,y) compared to the 2-spoke case, for both simulation andexperimental data (note reduced errors between middle and bottom row inFIG. 5).

FIG. 6 shows the final flip angle distributions for all methods. RFshimming generated a relatively homogeneous flip angle distribution (toprow in FIG. 6), matching the B₁ ⁺ distribution (top row in FIG. 4).Quantitatively, when targeting a homogeneous flip angle distribution,the standard RF-shimming technique yielded σ=6.4%, 13.0%, and 13.2% forsimulations, phantom, and human head experimental data, respectively.Fast-k_(z) yielded σ=6.1%, 12.3%, and 12.2% for simulations, phantom,and human head experimental data, respectively. The tailored excitationmethod yielded σ=6.3%, 11.8%, and 12.6% for simulations, phantom, andhuman head experimental data, respectively. Both fast-k_(z) and thetailored excitation method only marginally improved the flip anglehomogeneity comparing to single pulse using conventional RF shimming,since the assumption of B₁ ⁺ distribution for both methods were notsatisfied.

The combined SAGS and RF shimming approach also generated homogeneousflip angle distributions (middle and bottom rows in FIG. 6), even thoughtheir B₁ ⁺ distributions showed larger spatial variation than for RFshimming alone (FIG. 4 mid, bottom). Using combined RF shimming and 2pulse SAGS, we found σ=2.8%, 8.4%, and 9.2% for simulations, phantom,and human head experimental data, respectively. Using combined RFshimming and 4 pulse SAGS, we found σ=2.8%, 7.6%, and 7.9% forsimulations, phantom, and human head experimental data, respectively.Compared to RF shimming alone, the combined SAGS and RF shimmingapproach can generate a more homogeneous flip angle distribution. Thissuggests that, while a B₁ ⁺ distribution can be visually suboptimal,when combined with quadratic SEMs, it may still result in a relativelyhomogeneous flip angle distribution if SEMs are optimally used. Thebottom panel of FIG. 6 shows the flip angle distribution using theB_(1shim) ⁺ obtained from the combined SAGS and RF shimming method with99 equi-spaced spokes. The results are similar to those of the 4-spokecombined SAGS and RF shimming method, suggesting that the number ofspokes, after careful tuning of locations, amplitudes, and phases, isnot the bottleneck for further improving the flip angle homogeneity.

The initial guess for B_(1shim) ⁺(x,y) in the calculations for FIGS. 2and 4 was chosen as the CP mode of all transmission coils. We repeatedthe simulations using the result from conventional RF shimming as theinitial guess to test the sensitivity of our method. We found that theresults of the 2-spoke combined SAGS and RF shimming were not sensitiveto the initial guess for B_(1shim) ⁺(x,y) (FIG. 7). In fact, the flipangle distributions were similar when using either the CP mode (FIG. 6)or the B₁ ⁺ obtained with conventional RF shimming (FIG. 7) as initialguess.

Under the small flip angle approximation (1), we proposed a method toimprove flip angle homogeneity by remapping the spatial distribution ofB₁ ⁺ into a lower dimension coordinate system spanned by the nonlinearSEMs (SAGS) and optimizing a time-invariant combination of transmitamplitudes and phases from multiple RF coils (i.e., RF shimming) (Eq.[5]). In previous work on SAGS, we clearly demonstrated the advantage ofB₁ ⁺ remapping by implementing a simple pulse design using time-varyinglinear and nonlinear SEMs in a one-dimensional k-space to achieve ahomogenous flip angle distribution efficiently (6). This study furtherextends the advantage of B₁ ⁺ remapping to encompass RF shimming.Specifically, we used both simulated (20-channel transmit array) andexperimentally measured (8-channel transmit array) transmitsensitivities to demonstrate the benefits of using linear and quadraticSEMs to achieve k-space dimension reduction and ultimately improve thehomogeneity of the flip angle distribution (FIG. 6). Compared to theconventional one-spoke RF shimming method, the two-spoke SAGS method canimprove the homogeneity of flip angle by 44% (6.4%→2.8%) relativestandard deviation using simulation data, 47% (13.0%→6.1%) using phantomdata and 60% (13.2%→7.9%) using human head data (FIG. 6). Importantly,our two-spoke SAGS method also outperformed the two-spoke tailorexcitation method and 5-spoke fast-k_(z) method (FIG. 6).

We want to emphasize the importance of simultaneously optimizing thecombination of transmit coils and tailoring the combinations of linearand nonlinear SEMs. Intuitively, we could first use conventional RFshimming to search for the optimal complex-valued coil combinationcoefficients that achieve a homogeneous B₁ ⁺ distribution (16). Then wewould remap this B₁ ⁺ distribution into a lower dimension usingnonlinear SEMs. However, such a sequential approach may yieldsub-optimal results and it would be instead preferable to perform RFshimming and B₁ ⁺ remapping simultaneously, as suggested by the maps inthe top panel of FIG. 2.

Two current limitations of our approach for improving flip anglehomogeneity are related to B₁ ⁺ remapping accuracy M_(xy)(x,y)/B₁⁺(x,y)≈{tilde over (M)}_(xy)(h(x,y)) and to using a finite number ofspokes to construct a desired flip angle distribution

$\begin{matrix}{{{\overset{\sim}{M}}_{xy}\left( {h\left( {x,y} \right)} \right)} = {\sum\limits_{spokes}{{W_{F^{\prime}}\left( k_{h} \right)}{{\exp \left\lbrack {2\pi \; j\; {k_{h} \cdot {h\left( {x,y} \right)}}} \right\rbrack}.}}}} & \left( {{Eq}.\mspace{14mu} \lbrack 3\rbrack} \right)\end{matrix}$

In FIG. 3, we show that flip angle homogeneity is similar between theresults using 99 equi-spaced spokes and the results using 4 tailoredspokes. This suggests that using only 4 spokes is not the bottleneck inimproving the flip angle homogeneity, but rather that B₁ ⁺ remapping isthe key to the ultimate flip angle distribution. And comparing FIG. 5and FIG. 6, we can observe roughly the smaller the fitting error in FIG.5 corresponds to more homogeneous M_(xy) distribution in FIG. 6.

The pulses designed by our proposed method can be seen as special casesof combining pTx and nonlinear SEMs (19). However, there are somedifferences: 1) like the RF shimming method, we only need one commondriving RF amplifier to implement the RF pulse using a vector modulatorto deliver the same waveform with varying amplitudes and phases for eachtransmit coil (7). Thus, the complexity and the cost are less than thoseof a pTx system and nonlinear SEMs. 2) The pulse sequence involvingnonlinear SEMs is typically designed on a multi-dimensional k-space,whose dimension equals to the number of SEMs. SAGS-shim is one method toachieve similar results on a k-space with a reduced dimension. Thebenefit of such dimension reduction has been reported in our previouswork (6). In short, SAGS simplifies the pulse design by transforming theoptimization problem from a higher dimensional k-space to a lowerdimensional one by seeking appropriate combinations of SEMs.

In this invention, we only restricted the spoke locations to besymmetrically located around the center of k-space. Naturally, allowingspokes with arbitrary locations, amplitudes, and phases can improve theresults by increasing the degree of freedom of pulse design at the costof higher complexity in optimization. However, it should be noted that,even using 4 spokes with the restriction of conjugate locations andequal amplitudes, we already got reasonably homogeneous M_(xy)distribution (FIG. 6). Further improvement in M_(xy) homogeneity isexpected to be marginal by using more spokes or allowing a higher degreeof freedom in pulse design.

In conclusion, we proposed a combined SAGS and RF shimming approach tomitigate B₁ ⁺ inhomogeneity, a prominent artifact in high field imaging.Our simulations and experimental results suggest that this approach canbe one method to facilitate structural and functional imaging at highfields.

What is claimed is:
 1. A method for generating a flip angle spatialdistribution of magnetic resonance imaging, comprising: incorporatingthe usage of one or a plurality of spatial encoding magnetic fields andthe usage of a plurality of RF coils to generate a specific spatialdistribution of flip angle, wherein the plurality of RF coils excite aB₁ ⁺ field jointly, the ratio of signal amplitudes of the plurality ofRF coils is kept invariant during the excitation process, and the phaserelationship of signals from the plurality of RF coils is kept invariantduring the excitation process.
 2. The method as recited in claim 1,wherein the plurality of spatial encoding magnetic fields include linearand nonlinear spatial encoding magnetic fields.
 3. The method as recitedin claim 1, wherein the generated spatial distribution of flip angle issubstantially homogeneous.
 4. The method as recited in claim 1, furthercomprising adjusting the signal amplitudes of the plurality of RF coilsby a single controller, wherein the ratio of signal amplitudes and thephase relationship of signals of the plurality of RF coils are keptinvariant during the excitation process.
 5. An apparatus for generatinga flip angle spatial distribution of magnetic resonance imaging,comprising: one or a plurality of spatial encoding magnetic field coils;and a plurality of RF coils incorporated with the one or plurality ofspatial encoding magnetic field coils to generate a specific spatialdistribution of flip angle, wherein the plurality of RF coils excite aB₁ ⁺ field jointly, the ratio of signal amplitudes of the plurality ofRF coils is kept invariant during the excitation process, and the phaserelationship of signals from the plurality of RF coils is kept invariantduring the excitation process.
 6. The apparatus as recited in claim 6,wherein the plurality of spatial encoding magnetic field coils includelinear and nonlinear spatial encoding magnetic field coils.
 7. Theapparatus as recited in claim 6, wherein the generated spatialdistribution of flip angle is substantially homogeneous.
 8. Theapparatus as recited in claim 6, further comprising: a singlecontroller, adjusting the signal amplitudes of the plurality of RFcoils, wherein the ratio of signal amplitudes and the phase relationshipof signals of the plurality of RF coils are kept invariant during theexcitation process.